Flag nested-set complex real-rootedness conjecture

Let $\mathcal{L}$ be a geometric lattice and let $\mathcal{G}$ be a building set. The nested set complex $\mathbf{N}(\mathcal{L},\mathcal{G})$ is a simplicial complex, and being flag means that every minimal nonface has cardinality two.

Flag nested-set real-rootedness conjecture. If $\mathbf{N}(\mathcal{L},\mathcal{G})$ is flag, then its $h$-polynomial is real-rooted.

This conjecture is proposed as an interpolation between Bóna's real-rootedness theorem for second Eulerian polynomials and the conjecture for maximal building sets. It seeks a condition on the building set that is weaker than requiring the minimal building set but strong enough to ensure real-rootedness.